Given a bounded simply connected domain U ⊂ C having a Lyapunov curve as its boundary, let L(L2(U)) stand for the C*-algebra of all bounded linear operators acting on the Hilbert space L2(U) with Lebesgue area measure. We show that the smallest C*-subalgebra A of L(L2(U)) containing the singular integral operator (Formula Presented.) along with its adjoint (Formula Presented.) all multiplication operators a ∈ C(U¯), and all compact operators on L2(U), is spectrally regular. Roughly speaking the latter means the following: if the contour integral of the logarithmic derivative of an analytic C*-valued function f is vanishing (or is quasi-nilpotent), then f takes invertible values on the inner domain of the contour in question.

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Keywords Analytic vector-valued function, C*-algebra, Logarithmic residue, Spectral regularity, Two-dimensional singular integral operator
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Series Operator Theory: Advances and Applications
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Bart, H, Ehrhardt, T, & Silbermann, B. (2018). Spectral regularity of a C*-algebra generated by two-dimensional singular integral operators. In Operator Theory: Advances and Applications. doi:10.1007/978-3-319-75996-8_3